Protein Geometry Calculation: Improving Voronoi Accuracy

Due to the current state of technology, we are unable to resolve the exact physical nature of atomic structure. However, estimates are getting closer and closer, and we are on the verge of technological revolution. The importance of protein geometry analysis stems from its ability to predict protein function, and the consequences of various protein confirmations. The description of globular protein structure can lead to great incite about the function of the protein. The established method for calculation of protein density is the use of Voronoi polyhedra around atomic groups. The calculation of Voronoi polyhedra is used to define the relationships between adjacent atoms.

In this approach, each individual atom has its own convex polyhedron surrounding it, and the faces of the polyhedron is made up of all the dividing planes perpendicular to vectors connecting the atoms, and the edges are the intersections of those planes(1). An individual atom’s polyhedron is defined such that all points within the polyhedron are closer to this atom than any others(1). Points equidistant from two atoms lie on a dividing plane, points equidistant from three atoms lie on a line, and points equidistant from four atoms form a vertex(1). The system of calculation of protein volume incorporates the use of the Voroni volumes to find all vertices of the polyhedron around each atom, and collect these vertices to create the minimally sized polyhedron around each atom(2). The possible vertices can be found by solving the equation of a sphere using the coordinates of four atoms(1).

However, the atoms of the protein molecule are in constant flux, and the sizes of the atoms can vary, which is not taken into account using only the Voronoi polyhedral calculations(3). An average of the positions may be more useful, because the fluctuations of atomic positions can have effects chemically and biologically. In calculating the packing density in protein structures, the van der Waals radii of the atomic groups must be used in the calculation of the Voronoi polyhedra. Originally, the Voronoi procedure placed planes in the midpoint of the lines between neighboring atoms, which allocates the space to the constituent atoms, but this fails to take into account the fact that different atoms have different radii(3). Part of the van der Waals envelope of the larger atom gets allocated to the smaller atom. This is counteracted by the placing of the plane not at the midpoint, but a position proportional to the radii of both atoms(3). This can introduce small errors, however. A derivative of the Voronoi construction, called the Delaunay triangulation, is the natural method for determining packing neighbors(1).

This construction consists of lines perpendicular to the Voronoi faces that connect each pair of atoms that share a face. This is useful when trying to find the atoms nearest another, or when trying to find empty areas in the protein packing. Packing efficiency is defined as the volume of the object divided by the space it occupies. This numerical value is important in the comparison of equivalent atoms in different parts of a protein structure(1). In protein packing, this corresponds to the van der Waals volume divided by the Voronoi volume. In order to make such a calculation, the van der Waals radii must be known, but as yet there is no universal agreement about how one knows this value, especially for water molecules and polar atoms(3). It has been shown that packing of proteins is often very tight, and calculations showing 74% of the volume filled for close packed spheres(1). Voronoi volumes do not always give an accurate measure of the packing, and the spheres in some packing arrangements occupy as much as 75.5% of their Voronoi volumes(4).

In tight packing, small changes become very significant, and the use of a revised Voronoi system becomes advisable. Thomas Hales devised a system that would attribute more of the unaccounted space to the appropriate atom. His system is called “star decomposition”(4). Essentially, a star is a modified Voronoi cell with a batch of tetrahedral protuberances(4). These protuberances can account for more area than the polyhedra of the traditional system. Although the Voronoi polyhedra can give a reasonable account, the “star decomposition” system can improve accuracy in the calculation of volume.

The Voronoi construction has been important in the development of protein geometry analysis, and is a useful weighting factor for gaining understanding of volume and surface properties of proteins. The revised Voronoi method may be able to go even further in developing our understanding of molecular protein properties and the exact physical nature of atomic structure. By understanding the relationships of atoms in proteins, we may be able to gain deeper understanding of how this affects protein function.

References:

1. M Gerstein & F M Richards, "Protein Geometry: Volumes, Areas, and

Distances," (2000) chapter 22 of volume F of the International Tables for

Crystallography ("Molecular Geometry and Features" in "Macromolecular

Crystallography")

2. F M Richards (1977). Areas, Volumes, Packing, and Protein Structure. Ann. Rev.

Biophys. Bioeng. 6, 151-76.

3. J Tsai, R Taylor, C Chothia & M Gerstein (1999). "The Packing Density in Proteins:

Standard Radii and Volumes," J. Mol. Biol. 290: 253-266.

4. Barry Cipra (1998). “Packing Challenge Mastered At Last,” Science 281: 1267.